A Non-parametric Method for the Inference of Halo Occupation Distributions

1. Problem Statement

The relationship between galaxies and their host dark matter haloes, described by the Halo Occupation Distribution (HOD), is crucial for understanding galaxy formation and cosmological inference. Traditionally, HODs are inferred from the galaxy two-point correlation function (2PCF) using parametric models (e.g., the five-parameter model by Zheng et al. 2005).

Key Limitations of Parametric Approaches:

• Rigidity: Parametric models assume specific functional forms (e.g., smoothed step-functions for centrals, power laws for satellites). This restricts the space of possible HODs and can bias results if the true HOD deviates from the assumed form.
• Selection Bias: Different galaxy selection criteria (e.g., stellar mass $M_*$ vs. star formation rate SFR) produce distinct HOD shapes that standard parameterizations often fail to capture accurately.
• Computational Cost: Inferring HODs via Markov Chain Monte Carlo (MCMC) requires forward-modeling the 2PCF at every step. Calculating the 2PCF for large galaxy samples is computationally expensive ($O(N \log N)$), making non-parametric inference (which requires exploring a vast, unconstrained space) infeasible with brute-force methods.

2. Methodology

The authors propose a non-parametric HOD inference framework that avoids assuming a functional form for the HOD. Instead, it treats the physical selection thresholds (stellar mass and SFR) as parameters and uses machine learning to map these to the HOD and 2PCF.

A. Data and Simulations

• Training Data: The emulator is trained on lightcones generated using the Santa Cruz Semi-Analytic Model (SC SAM) applied to the SMDPL dark matter-only $N$-body simulation.
• Test Data: The method is validated on SC SAM mocks and tested on the IllustrisTNG100-1 hydrodynamic simulation to assess robustness against different physics implementations.
• Selection Criteria: Galaxy samples are selected based on lower-bound thresholds for stellar mass ($M_*$) and instantaneous star formation rate (SFR).

B. The Emulator (Gaussian Processes)

To bypass the computational cost of calculating the 2PCF directly, the authors train a Gaussian Process (GP) emulator.

• Inputs: Physical galaxy property thresholds ($\log M_*$, $\log \text{SFR}$).
• Outputs: Three histograms that fully characterize the clustering and occupation:
  1. Normalized one-halo pair counts ($dd_{1h}$).
  2. Central galaxy number density vs. halo mass ($n_{cen}$).
  3. Satellite galaxy number density vs. halo mass ($n_{sat}$).
• 2PCF Reconstruction: The emulator predicts the above histograms, which are then combined with an analytical approximation for the two-halo term (using halo bias and linear dark matter correlation) to reconstruct the angular two-point correlation function, $w(\theta)$.
• Efficiency: This approach reduces the time for a single likelihood evaluation by a factor of >500 compared to brute-force pair counting.

C. Inference Framework

The authors implement a Bayesian MCMC framework (using emcee) where:

• The likelihood compares the observed $w(\theta)$ (mocked or real) to the emulator-predicted $w(\theta)$.
• The parameters being inferred are the physical selection thresholds ($M_*$, SFR) and the resulting HODs.
• Priors: Uniform priors are placed on the selection thresholds, and a Gaussian prior is placed on the galaxy number density to break degeneracies.

3. Key Contributions

1. Non-Parametric HOD Inference: A novel framework that infers HODs without assuming a specific functional form, allowing the data to dictate the shape of the central and satellite occupation distributions.
2. Emulator-Accelerated Likelihood: The development of a GP-based emulator that maps physical selection criteria to HODs and 2PCFs, making non-parametric inference computationally feasible.
3. Joint Inference: The ability to simultaneously infer the physical selection limits (e.g., the effective $M_*$ and SFR cuts) and the HOD from a single 2PCF measurement.
4. Benchmarking: A rigorous comparison showing that this non-parametric approach outperforms or matches standard parametric models (Zheng et al. 2005; Geach et al. 2012) in accuracy and precision, particularly for complex selection criteria like SFR.

4. Results

Validation on SC SAM (Training Set)

• The emulator achieves high fidelity, with median fractional residuals < 1% for the underlying histograms and < 5% for the reconstructed $w(\theta)$.
• Recovery: The framework recovers the "true" HODs (central, satellite, and total) within 0.1 dex for the SC SAM validation set.
• Thresholds: It successfully recovers the input $M_*$ and SFR thresholds with high precision (median arithmetic residuals $\approx 0.01$ dex).

Testing on TNG100-1 (Independent Simulation)

• Central HOD: Recovered consistently within 0.2 dex.
• Satellite HOD: The method tends to underestimate the satellite HOD by $\gtrsim 0.3$ dex, particularly at lower halo masses.
  • Cause: This discrepancy arises from differences in the SC SAM and TNG100-1 simulations regarding satellite radial distributions, quenching mechanisms, and satellite multiplicities. The emulator, trained on SC SAM physics, learns a specific $w(\theta)$-HOD mapping that does not perfectly translate to TNG100-1 physics.
• Comparison to Parametric Models: When compared to standard parametric fits on TNG100-1 mocks, the non-parametric approach consistently yields comparable or better precision and accuracy. Parametric models often fail to capture the unique shapes of SFR-selected HODs, leading to larger residuals and biased uncertainties.

5. Significance and Future Directions

• Cosmological Readiness: As next-generation surveys (DESI, Euclid, Rubin, Roman) probe diverse galaxy populations and redshifts, rigid parametric models will become a bottleneck. This flexible, non-parametric approach is essential for capturing the complexity of real galaxy populations without introducing model bias.
• Computational Feasibility: By accelerating likelihood evaluations, this method opens the door to complex, high-dimensional HOD inference that was previously computationally prohibitive.
• Future Work:
  • Multi-Simulation Training: Training the emulator on a suite of simulations (e.g., OpenUniverse, Horizon Run 5) to improve robustness against simulation-specific physics.
  • Real Data Application: Applying the method to real spectroscopic and photometric surveys, potentially incorporating redshift-space distortions.
  • Simultaneous Inference: Extending the framework to simultaneously infer the HOD and the underlying distribution of physical properties corresponding to observed flux limits.

In conclusion, this paper presents a robust, machine-learning-enhanced framework that overcomes the limitations of traditional parametric HOD modeling, offering a more accurate and flexible tool for interpreting galaxy clustering in the era of precision cosmology.